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In this paper we provide a complementary set of results to those we presentin our companion work \cite{Stojnicl1HidParasymldp} regarding the behavior ofthe so-called partial $\ell_1$ (a variant of the standard $\ell_1$ heuristicoften employed for solving under-determined systems of linear equations). As iswell known through our earlier works\cite{StojnicICASSP10knownsupp,StojnicTowBettCompSens13}, the partial $\ell_1$also exhibits the phase-transition (PT) phenomenon, discovered and wellunderstood in the context of the standard $\ell_1$ through Donoho's and our ownworks \cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}.\cite{Stojnicl1HidParasymldp} goes much further though and, in addition to thedetermination of the partial $\ell_1$'s phase-transition curves (PT curves)(which had already been done in\cite{StojnicICASSP10knownsupp,StojnicTowBettCompSens13}), provides asubstantially deeper understanding of the PT phenomena through a study of theunderlying large deviations principles (LDPs). As the PT and LDP phenomena areby their definitions related to large dimensional settings, both sets of ourworks, \cite{StojnicICASSP10knownsupp,StojnicTowBettCompSens13} and\cite{Stojnicl1HidParasymldp}, consider what is typically called the asymptoticregime. In this paper we move things in a different direction and considerfinite dimensional scenarios. Basically, we provide explicit performancecharacterizations for any given collection of systems/parameters dimensions. Wedo so for two different variants of the partial $\ell_1$, one that we callexactly the partial $\ell_1$ and another one, possibly a bit more practical,that we call the hidden partial $\ell_1$.

Partial $\ell_1$ optimization in random linear systems -- finite dimensions